Exact scaling exponents in Korn and Korn-type inequalities for cylindrical shells

Understanding asymptotics of gradient components in relation to the symmetrized gradient is im- portant for the analysis of buckling of slender structures. For circular cylindrical shells we obtain the exact scaling exponent of the Korn constant as a function of shell's thickness. Equally sharp results are obtained for individual components of the gradient in cylindrical coordinates. We also derive an ana- logue of the Kirchho? ansatz, whose most prominent feature is its singular dependence on the slenderness parameter, in marked contrast with the classical case of plates and rods

Gaussian curvature as an identifier of shell rigidity

In the paper we deal with shells with non-zero Gaussian curvature. We derive sharp Korn's first (linear geometric rigidity estimate) and second inequalities on that kind of shells for zero or periodic Dirichlet, Neumann, and Robin type boundary conditions. We prove that if the Gaussian curvature is positive, then the optimal constant in the first Korn inequality scales like $h,$ and if the Gaussian curvature is negative, then the Korn constant scales like $h^{4/3},$ where $h$ is the thickness of the shell. These results have classical flavour in continuum mechanics, in particular shell theory. The Korn first inequalities are the linear version of the famous geometric rigidity estimate by Friesecke, James and M\"uller for plates [14] (where they show that the Korn constant in the nonlinear Korn's first inequality scales like $h^2$), extended to shells with nonzero curvature. We also recover the uniform Korn-Poincar\'e inequality proven for "boundary-less" shells by Lewicka and M\"uller in [37] in the setting of our problem. The new estimates can also be applied to find the scaling law for the critical buckling load of the shell under in-plane loads as well as to derive energy scaling laws in the pre-buckled regime. The exponents $1$ and $4/3$ in the present work appear for the first time in any sharp geometric rigidity estimate.Comment: 25 page

Homogenization via unfolding in periodic layer with contact

In this work we consider the elasticity problem for two domains separated by a heterogeneous layer. The layer has an $\varepsilon-$periodic structure, $\varepsilon\ll1$, including a multiple micro-contact between the structural components. The components are surrounded by cracks and can have rigid displacements. The contacts are described by the Signorini and Tresca-friction conditions. In order to obtain preliminary estimates modification of the Korn inequality for the $\varepsilon-$dependent periodic layer is performed. An asymptotic analysis with respect to $\varepsilon \to 0$ is provided and the limit problem is obtained, which consists of the elasticity problem together with the transmission condition across the interface. The periodic unfolding method is used to study the limit behavior.Comment: 20 pages, 1 figur

Rigorous derivation of the formula for the buckling load in axially compressed circular cylindrical shells

The goal of this paper is to apply the recently developed theory of buckling of arbitrary slender bodies to a tractable yet non-trivial example of buckling in axially compressed circular cylindrical shells, regarded as three-dimensional hyperelastic bodies. The theory is based on a mathematically rigorous asymptotic analysis of the second variation of 3D, fully nonlinear elastic energy, as the shell's slenderness parameter goes to zero. Our main results are a rigorous proof of the classical formula for buckling load and the explicit expressions for the relative amplitudes of displacement components in single Fourier harmonics buckling modes, whose wave numbers are described by Koiter's circle. This work is also a part of an effort to quantify the sensitivity of the buckling load of axially compressed cylindrical shells to imperfections of load and shape